Circumference

Circumference length

Wrap a string around a can and then stretch it over the diameter: it fits just over three times. This goes for any can, wheel or coin. This “just over three” is the number π. Who discovered That the outline of a circle is a little larger than three diameters was already known by the Babylonians and Egyptians almost 4 000 years ago; they used values like 3, $3\tfrac{1}{8}$ and 3,16. The first calculated π with certainty was Archimedes of Syracuse (3rd century BC): using polygons with 96 sides, he proved that π is between $3\tfrac{10}{71}$ and $3\tfrac{1}{7}$. The symbol π was used by the Welshman William Jones in 1706 and popularized by Leonhard Euler.

October 12, 2026 Read

Chapter exercises 10 — Circumference and circle

Exercises and resolutions 336 Determine the radius of the center circle $O$, given $AB = 3x - 3$ and $OA = x + 3$. Ver resposta$r = 12$ 337 The circle has a radius of 16 cm and the point $P$ is 7 cm from the center. Determine the distance between $P$ and the circumference. Ver resposta9 cm

October 11, 2026 Read

Tangent segments — circumscribable quadrilaterals

From a point outside a circle two tangents come out, and the two segments to the points of contact have the same length. Who discovered The equality of tangent segments is a direct consequence of proposition III.17 of The Elements of Euclid, which teaches how to draw the tangent through an exterior point. This gives rise to Pitot’s theorem: in a quadrilateral that has an inscribed circle, the sums of the opposite sides are equal. The French engineer Henri Pitot published it in 1725 (see the curiosity about him below). The reciprocal — if the sums of the opposite sides are equal, the quadrilateral has an inscribed circumference — was proven in the 19th century and is usually attributed to Jakob Steiner.

October 11, 2026 Read

Relative positions of two circles

Two circles can be secant, tangent, one inside the other or far apart. The distance between the centers decides which is the case. Who discovered The basic cases are in The Elements of Euclid (c. 300 BC): two distinct circles intersect at a maximum of two points (proposition III.10), and, when they touch, the point of contact is on the straight line that joins the centers (III.11 and III.12). A century later, Apollonius of Perga proposed the most famous problem on the subject: given three circles, draw a fourth tangent to all of them.

October 11, 2026 Read

Relative positions of straight line and circle

A straight line can cut a circle at two points, touch it at just one or pass far. The line that only touches is the tangent. Who discovered The tangent properties are in book III of The Elements by Euclid (c. 300 BC). The proposition III.16 shows that the straight line perpendicular to the radius, at its extremity, touches the circle at a single point, and the III.18 proves the opposite: the tangent is perpendicular to the radius at the point of contact. The word tangent comes from the Latin tangere, “to touch”.

October 11, 2026 Read

Definitions and elements

All points at the same distance from a center: the definition of the circle fits on a line, and yet it hides the number π. Who discovered The circle is one of the oldest forms in human history, from the wheels of Mesopotamia (c. 3500 B.C.) to stone monuments. The rigorous definition is from Euclid (definitions I.15 to I.18 of The Elements): a figure limited by a line whose points are all at the same distance from an interior point, the center. His third postulate guarantees that it is possible to draw a circle with any center and any radius. The first person to accurately calculate the ratio between contour and diameter was Archimedes (3rd century BC).

October 11, 2026 Read

Circumference ratio in regular polygons

In previous posts we talked about Stewart’s theorem, here we will talk about the circumference relationship in regular polygons. The relationship between circles and regular polygons is a fascinating and complex topic in geometry. Regular polygons are geometric figures with congruent sides and angles, and the circle can be seen as a limiting case of a regular polygon with an infinite number of sides. When a regular polygon is inscribed in a circle, each of its vertices touches the circle, and the center of the polygon coincides with the center of the circle. This allows precise relationships to be established between the radius of the circle and the dimensions of the polygon, such as the side and apothem (the distance from the center to a side).

July 18, 2024 Read

Stewart Relationship

In previous posts we talked about ptheorems for any triangles. The Stewart Relation, also known as Stewart’s Theorem, is an important formula in geometry that relates the lengths of the sides of a triangle to the length of a cevian. A cevian is a line segment that starts from a vertex of a triangle and meets the opposite side. This relationship is useful for solving problems involving determining the lengths of segments within a triangle, such as medians, bisectors, and heights.

July 12, 2024 Read

Theorem for any triangles

In previous posts we talked about point power. The Sine Theorem is a fundamental mathematical tool in trigonometry, which establishes an important relationship between the sides of a triangle and the sines of its angles. This theorem is applicable to any type of triangle, be it acute, obtuse or scalene, and is essential for solving problems involving the geometry of triangles. The Law of Sines states that, in any triangle, the ratio between the length of one side and the sine of the opposite angle is constant and equal for all three sides and angles of the triangle.

July 12, 2024 Read

Point Power

In previous posts we talked about inscribed angle. The power of a point is a real number that reflects the relative distance of a given point to a given circle. To understand this better, imagine you have a point A outside a circle. Draw two secants of A that intersect the circle at two points each. The power of point A is then defined as the product of the distances from A to one point of intersection with the product of the distances from A to the other point of intersection on the other secant.

July 10, 2024 Read

Segment angle

In previous posts we talked about inscribed angle. The segment angle is defined as the angle formed between a chord of a circle and the tangent to the circle at one of the ends of the chord. This angle can be seen as a special case of the inscribed angle, where the vertex of the angle is precisely on one of the points that define the arc. To understand better, imagine a circle with a rope extended from one point to another. At one of the points where the rope touches the circle, draw a tangent line. The angle formed between this tangent and the chord is what we call the segment angle. An interesting property of this angle is that it is half the measure of the central angle that subtends the same arc.

July 9, 2024 Read

Inscribed angle

In previous posts we talked about properties of circular arcs. Now, we talk about angles inscribed in a circle. An inscribed angle is one whose vertex is on a circle and whose sides are secants that intersect the circle at two other distinct points. The most notable property of an inscribed angle is that its measure is equal to half the measure of the arc it subtends on the circumference.

July 4, 2024 Read